Questions tagged [proof-techniques]

Questions about general methods and techniques for proving multiple theorems. When asking about the proof of a single statement, use tags relating to what the proof is about instead.

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How to create mathematical proof of TSP and SLAP equivalence?

In my thesis, I'm dealing with SLAP (storage location assignment problem) -- which is finding optimal distribution of products to location slots in a generic warehouse. My aim was to implement ...
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Longest path length in an undirected tree, can we prove this algorithm is correct (which it is)?

Hello I solved this leetcode https://leetcode.com/problems/tree-diameter/ question reserved for people who pay the subscription. The question: Given an undirected tree (tree is not disjoint), ...
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1answer
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Need hint for bipartiteness proof

I am given a graph $G = (V, E)$ with $N$ connected components and $G^\prime = (V^\prime, E^\prime)$, where for each $v \in V$ there is $v_1, v_2 \in V^\prime$ and for each $(u, v) \in E$ there is $(...
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Why is Relativization a barrier?

When I was explaining the Baker-Gill-Solovay proof that there exists an oracle with which we can have, $\mathsf{P} = \mathsf{NP}$, and an oracle with which we can have $\mathsf{P} \neq \mathsf{NP}$ to ...
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reduction from ALLTM to ETM

I am trying to understand why this "proof" of ETM undecidability is wrong. ALLTM={ < M >|M is a TM, L(M)=∑*} ETM={< M >|M is a TM, L(M)=∅} We know that ALLTM is undecidable, lets ...
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Language of CFG: $S \to aS | aSbS | \varepsilon$

I'm trying to prove that the language L generated by the CFG $S \to aS | aSbS | \varepsilon$ is the language $L=\{ w \in \{a,b\}^*: \text{every prefix of $w$ has at least as many $a$'s as $b$'s} \}$.I ...
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Prove (p → ¬q) is equivalent to ¬(p ∧ q)

I need to prove the above sequent using natural deduction. I did the first half already i.e. I proved $(p\rightarrow\neg q)\rightarrow \neg (p \wedge q)$, but I'm stuck on where to start for the ...
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27 views

Time complexity of binary search

Proposition: The binary search algorithm runs in $O(\log n)$ time for a sorted sequence with $n$ elements. When justifying this claim, first we say that with each recursive call the number of ...
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1answer
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Early Deadline First (EDF) scheduling in real-time systems feasibility test proof

I am trying to prove Theorem 6.2 on page 127 of the book Real-Time Systems by Jane W. S. Liu: http://www.cse.hcmut.edu.vn/~thai/books/2000%20_%20Liu-%20Real%20Time%20Systems.pdf It is based on Early ...
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How to prove a statement in regular expression?

I cannot figure out how to go about proving this statement in regular expression. $$ L(R_1) \subseteq L(R_2) \subseteq L(R_3) \implies L(R_1^*+R_3)^* \subseteq L(R_2^*+R_3^*) $$ Here's what I have ...
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Having trouble understanding how to prove a language context free? [duplicate]

I've been studying the theory of automata. I came across this problem in the book and unable to understand how to solve this. I've solved some examples using the Pumping lemma but this one uses ...
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Proving van Emde Boas recurrence

I have tried to solve the following question: van Emde Boas Bounds Show that $T(u) = T(\sqrt{u}) + O(1)$ has the solution $T(u) = O(\log\log u)$. Hint: consider the binary representation of $u$. ...
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When are existential quantifiers in the intuitionistic propositional calculus eliminable and when not

I am so ignorant I don't even know where should I ask this - on FOM? On mathoverflow? On cstheory? So please consider as sort of a meta-question readdressing me in case you think this is a wrong site. ...
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1answer
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Why discriminate the base case allows me to complete the induction proof?

I have a successful completed proof which used induction. but I essentially proved the goal on the base case by tactic discriminate. Why is this induction proof ...
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How to prove greedy algorithm is correct

I have a greedy algorithm that I suspect might be correct, but I'm not sure. How do I check whether it is correct? What are the techniques to use for proving a greedy algorithm correct? Are there ...
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38 views

How would I prove that nondeterministic Turing machines are undecidable?

How would I go about proving that the language: $$A_{NTM }= \{\langle N, w\rangle | N \text{ is a nondeterministic TM and } N \text{ accepts }w\}$$ is undecidable? I looked at the proof for $A_{TM}$ ...
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Can every sentence of first-order logic be converted into an equisatisfiable equation in Boolean algebra?

There may be some theoretical literature, unknown to me, that addresses this question. If possible, I would like a practical approach to this problem. My attempt involves the use of an equational ...
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Why are mathematical proofs so hard?

I am an electrical engineer and trying to make a transition into machine learning. I read in multiple articles that I have to learn data structures and algorithms, before this I have to learn about ...
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1answer
86 views

On the proof techniques of Udi Manber

I was familiar with the approach of first coming up with an algorithm, and then proving the loop invariant to come up with an algorithm as elucidated in CLRS (Introduction to algorithms, Thomas H. ...
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1answer
30 views

Converse proof for random coding capacity of AVC

I want to see the converse proof for the random coding (shared randomness) capacity of AVC. All I can find online is Csiszar Narayan's AVC paper which looks at deterministic coding. Further, the proof ...
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Are different assignments allowed for the implication graph proof of 2-SAT being in P?

One proof for $2-SAT$ being in $P$ uses the implication graph, i.e. one creates 2 vertices per variable $a$, one for each possible literal ($a$ and $\neg a$). One then adds 2 arcs per clause $(a \lor ...
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why is behaviour of simpl differ so much after a commutative operation and how to inspect simpl?

In Coq, while trying to prove a lemma mult_n_Sm for mult_comm, I have this equation in a proof: ...
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1answer
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Why isn't plus_assoc rewriting correctly?

First I have plus_assoc ready. Theorem plus_assoc : forall n m p : nat, n + (m + p) = (n + m) + p. for simplicity we omit the ...
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Closed form of recurrence with two inputs

This question comes from a relatively simple coding challenge at Codesignal, but represents an interesting CS/math puzzle. The question states: "When a candle finishes burning it leaves a ...
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Is every unambiguous grammar regular?

While searching for an answer to this question I found out that there is an unambiguous grammar for every regular language. But is there a regular language for every unambiguous grammar? How can I ...
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67 views

Hardness of maximizing difference of functions

Suppose that the problem of maximizing a real function $f$ over a certain domain $D$ is NP_HARD. What can be said about the problem of maximizing $f-g$, with $g$ being another function over $D$? Is it ...
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1answer
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Double hashing constraints

For double hashing, we have some constraints on $h'(k)$ (1) It should never evaluate to 0 (2) It should be relatively prime to m How to show that all slots in an open addressing table will be ...
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Constant in Substitution method for recurrence

The solution for solving the following recurrence with the substitution method involves adding the a constant inside the recurrence, which is confusing to me. This is question 4.3-2 in the CLRS ...
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Proof using tableau algorithm

I have spent an hour finding the answer to this problem but can't do it, this is the problem: Determining whether the following semantic entailment holds or not by using the tableau algorithm ...
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1answer
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Why doesn't diagonalization require taking a limit?

When we quantify infinite sums, we do so by taking the limit as $i$ goes to infinity. For example, we look at $\lim_{n\rightarrow \infty}\sum_{n\in \mathbb{N}}n$, and then we say that this diverges ...
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what is the relevance of computability when applying diagonallization?

When thinking about diagonalization, I've always glossed over whether or not the enumeration, or the diagonal is computable or not. When does it matter? Say for example, that have an enumeration of ...
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An example of something you can formally verify with proofs in Software Development

I have been working on understanding formal verification of software. Formal methods include things like modeling your software with Petri Nets, Automata, or State-Transition Graphs. Other techniques ...
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1answer
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Proof for time complexity of Insertion (k-proximate) Sort equals O(nk)

The following is the definition for Proximate Sorting given in my paper: An array of distinct integers is k-proximate if every integer of the array is at most k places away from its place in the array ...
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Understanding the proof of “DFS of undirected graph $G$, yields either tree edge or back edge” better with graph for each statement in proof

I was going through the edge classification section by $\text{DFS}$ algorithm on an undirected graph from the text Introduction to Algorithms by Cormen et. al. where I came across the following proof. ...
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1answer
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Difficulty in understanding a portion in the proof of the $\text{“white path”}$ theorem as with in CLRS text

I was going through the $\text{DFS}$ section of the Introduction to Algorithms by Cormen et. al. and I faced difficulty in understanding the $\Leftarrow$ direction of the proof of the white path ...
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1answer
56 views

Difficulty in understanding a statement in the proof of the correctness of $\text{BFS}$ algorithm as dealt with in CLRS

I was going through section of Breadth First Search of the text Introduction to Algorithms by Cormen et. al. and I faced difficulty in understanding a statement in the proof below which I have marked ...
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1answer
55 views

Nondeterministic polynomial time algorithm versus certificate/verifier for showing membership in NP

In this paper (https://arxiv.org/pdf/1706.06708.pdf) the authors prove that optimally solving the $n\times n\times n$ Rubik's Cube is an NP-complete problem. In the process, they must show that the ...
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Difficulty in understanding the proof of the lemma : “Matroids exhibit the optimal-substructure property”

I was going through the text "Introduction to Algorithms" by Cormen et. al. where I came across a lemma in which I could not understand a vital step in the proof. Before going into the lemma ...
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1answer
49 views

Semi-decidability of the language $\overline{L_{\epsilon}}$

Firstly consider the problem: given $L_H = \{R(M)w : M \in TM_0, w\in L(M)\}$ where $R(M)$ are encoded transitions of $M \in TM_0$. Assume for contradiction $\overline{L_{H}}$ is semi-decidable, then ...
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1answer
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Pumping Lemma for CFL - $ \{ 0^{i} 1^{j} 0^{k} 1^{l} \hspace{0.2cm}| \hspace{0.2cm} i = l \hspace{0.2cm} \land j = k \} $

I was making exercices about the Pumping Lemma for CFL, and I stumbled up on this language: $$ \{ 0^{i} 1^{j} 0^{k} 1^{l} \hspace{0.2cm}| \hspace{0.2cm} i = l \hspace{0.2cm} \land j = k \} $$ I ...
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1answer
62 views

Recurrence relation for the number of “references” to two mutually recursive function

I was going through the Dynamic Programming section of Introduction to Algorithms (2nd Edition) by Cormen et. al. where I came across the following recurrence relations in the context of assembly line ...
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In Universal Hashing- probability of poor performance is small and is the same for any set of keys of the same size

I was going through the text Introduction to Algorithms by Cormen et. al. where I came across the following claim: Universal Hashing uses randomization.For the example of a compiler's symbol table, ...
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Difficulty in understanding few steps in the proof: “The class $\mathscr{H}_{p,m}$ of hash functions is universal”

I was going through the text Introduction to Algorithms by Cormen et. al. where I came across the following excerpt regarding the said proof and the steps where I felt difficulty are marked with $\...
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2answers
261 views

Induction proofs in Big-O notation

I'm not sure how go about this question: Prove the following inequality. For a correct proof, we require a value of the constant $c>0$ and an $n \in \mathbb N$, such that $\forall n>N : f(x)<...
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1answer
49 views

Clarification of the proof involving the regularity condition in Master Theorem

I was going the text Introduction to Algorithms by Cormen et al. Where I came across the following statement in the proof of the third case of the Master's Theorem. (The Statement of Master theorem) ...
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1answer
33 views

Language of lists of words, not all of which are different, is not context-free

How do I prove that the following language isn't context-free using the pumping lemma? $$ L=\{w_1\#w_2\#\dots\#w_k \colon k ≥ 2, w_i \in \{0,1\}^*, w_i = w_j \text{ for some } i \ne j\} $$ I am having ...
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1answer
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Proof that if P=PSPACE, RP=BPP

Like the title says. I can't figure out how to prove this. I think it probably has to do with the polynomial hierarchy collapsing but I'm not sure.
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Is the BPP class closed for union and intersection?

Just like the title says. I want to prove that given two languages $L_1,L_2 \in BPP$, $L_1 \cup L_2 \in BPP$ and $L_1 \cap L_2 \in BPP$

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